In this paper [properties of field functionals and characterization of local functionals][1] they have at page 5 
> Definition II.2. Let $U$ be an open subset of a Hausdorff locally convex space $E$ and let $f$ be a nap from $U$ to a Hausdorff locally convex space $F$. Then $f$ is said to have a derivative at $x \in U$ in the lirection of $v \in E$ if the following limit exists: ${ }^{.3}$
$$
D f_x(v):=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t} .
$$

>Definition II.3. Let $U$ be an open subset of a Hausdorff locally convex space E and let fe a map from $U$ to a Hausdorff locally convex space $F$. Then $f$ is Bastiani differentiable on $U$ [denoted by $\left.f \in C^1(U)\right]$ iff has a Gâteaux differential at every $x \in U$ and the map $D f: U \times E \rightarrow F$ defined by $D f(x, v)=D f_x(v)$ is continuous on $U \times E$.
 
at page  24 they have
> Lemma VI.2. Let $U$ be an open subset of $C^{\infty}(M)$ and $F: U \rightarrow \mathbb{K}$ be Bastiani smooth. For every $\varphi$ such that the distribution $D F_{\varphi} \in \mathcal{E}^{\prime}(M)$ has an empty wave front set, there exists a unique function $\nabla F_{\varphi} \in \mathcal{D}(M)$ such that
$$
D F_{\varphi}[h]=\int_M \nabla F_{\varphi}(x) h(x) d x
$$

Here $\mathcal{D}(M)$ is the space of test function on $M$ and $\mathcal{E}^{\prime}(M)$
is the dual of the space of section $\Gamma (E)$

Since $h(x) is a section in Lemma VI.2,  how can we produce a  number by multiplying  it by a function and then intergrate it ? 

Am I missing something?

  [1]: https://hal.science/hal-01654120/document